Anyway, Morgenstern illustrated the problem with a scenario from The Adventures of Sherlock Holmes. In the story The Final Problem, Holmes was attempting to elude Professor Moriarty while traveling from London to Paris. It wasn't obvious that Holmes could simply outthink Moriarty. Moriarty might anticipate what Holmes was thinking. But then Holmes could anticipate Moriarty's anticipation, and so on: I think that he thinks that I think that he thinks, ad infinitum, or at least nauseum.10 Consequently, Morgenstern concluded, the situation called for strategy. He returned to the Holmes–Moriarty issue in a 1935 paper exploring the paradoxes of perfect future knowledge.
At that time, after a lecture on these issues, a mathematician named Eduard Cech approached Morgenstern and told him about similar ideas in von Neumann's 1928 paper on parlor games. Morgenstern was entranced, and he awaited an opportunity to meet von Neumann and discuss the relevance of the 1928 paper to Morgenstern's views on economics.
The chance came in 1938, when Morgenstern accepted a three-year appointment to lecture at Princeton University. (Von Neumann had by then taken up his position at the nearby Institute for Advanced Study.) "The principal reason for my wanting to go to Princeton," Morgenstern said, "was the possibility that I might become acquainted with von Neumann."11 As Morgenstern told the story, he soon revived von Neumann's interest in game theory and began writing a paper to show its relevance to economics. As von Neumann critiqued early drafts, the paper grew longer, with von Neumann eventually joining Morgenstern as a coauthor. By this time—it was now 1940—the paper had grown substantially, and it kept growing, ultimately into a book published by the Princeton University Press in 1944. (Subsequent historical study suggests, though, that von Neumann had previously written most of the book without Morgenstern's help.12)
Theory of Games and Economic Behavior instantly became the game theory bible. In the eyes of game theory believers, it was to economics what Newton's Principia was to physics. It was a sort of newtonizing of Adam Smith, providing mathematical rigor to describe how individual interactions affect a collective economy. "We hope to establish," wrote von Neumann and Morgenstern, "that the typical problems of economic behavior become strictly identical with the mathematical notions of suitable games of strategy." It will become apparent, they asserted, that "this theory of games of strategy is the proper instrument with which to develop a theory of economic behavior."13 The authors then developed the theory throughout more than 600 pages, dense with equations and diagrams. But the opening sections are remarkably readable, laying out the authors' goals and intentions in a kind of extended preamble designed to persuade skeptical economists that their field needed an overhaul.
While noting that many economists had already been using mathematics, von Neumann and Morgenstern declared that "its use has not been highly successful," especially when compared to other sciences such as physics. Throughout its early pages, the book draws on physics as the model for how math can make murky knowledge precise and practical—in contrast to economics, where the basic ideas had been expressed so fuzzily that past efforts to use math had been doomed. "Economic problems … are often stated in such vague terms as to make mathematical treatment a priori appear hopeless because it is quite uncertain what the problems really are," the authors wrote.14 What economics needed was a theory that made precise and meaningful measurements possible, and game theory filled the bill.
Von Neumann and Morgenstern were careful to emphasize, though, that their theory was just a first step. "There exists at present no universal system of economic theory," they wrote, and if such a theory were ever to be developed, "it will very probably not be during our lifetime."15 But game theory could provide the foundation for such a theory, by focusing on the simplest of economic interactions as a guide to developing general principles that would someday be able to solve more complicated problems. Just as modern physics began when Galileo studied the rather simple problem of falling bodies, economics could benefit from a similar understanding of simple economic behavior.
"The great progress in every science came when, in the study of problems which were modest as compared with ultimate aims, methods were developed that could be extended further and further," von Neumann and Morgenstern declared.16 And so it made sense to focus on the simplest aspect of economics—the economic interaction of individual buyers and sellers. While economic science as a whole involves the entire complicated system of producing and pricing goods, and earning and spending money, at the root of it all is the choicemaking of the individuals participating in the economy.
ROBINSON CRUSOE MEETS GILLIGAN
Back in the days when von Neumann and Morgenstern were working all this out, standard economic textbooks extolled a simple economic model of their own, called the "Robinson Crusoe" economy. Stranded on a desert island, Crusoe was an economy unto himself. He made choices about how to use the resources available to him to maximize his utility, coping only with the circumstances established by nature.
Samuel Bowles, an economist at the University of Massachusetts, explained to me that textbooks viewed economics as just the activities of many individual Robinson Crusoes. Where Crusoe interacted with nature, consumers in a big-time economy interacted with prices. And that was the standard "neoclassical" view of economic theory. "That's what everybody taught," Bowles said. "But there was something odd about it." It seemed to be a theory of social interactions based on someone who had interacted only nonsocially, that is, with nature, not with other people. "Game theory adopts a different framework," Bowles said. "I'm in a situation in which my well-being depends on what somebody else does, and your well-being depends on what I do—therefore we are going to think strategically."17
And that's exactly the point that von Neumann and Morgenstern stressed back in 1944. The Robinson Crusoe economy is fundamentally different, conceptually, from a Gilligan's Island economy. It's not just the complication of social influences from other people affecting your choices about the prices of goods and services. The results of your choices—and thus your ability to achieve your desired utility—are inevitably intertwined with the choices of the others. "If two or more persons exchange goods with each other, then the result for each one will depend in general not merely upon his own actions but on those of the others as well," von Neumann and Morgenstern declared.18
Mathematically, that meant that no longer could you simply compute a single simple maximum utility for Robinson Crusoe. Your calculations had to accommodate a mixture of competing goals, maximum utilities for Gilligan, the Skipper too, the millionaire, and his wife, the movie star, the Professor and Mary Ann. "This kind of problem is nowhere dealt with in classical mathematics," von Neumann and Morgenstern noted.
Indeed, Bentham's notion of the "greatest possible good of the greatest possible number" is mathematically meaningless. It's like saying you want the most possible food at the least possible cost. Think about it—you can have zero cost (and no food) or all the food in the world, at a very high cost. Which do you want? You certainly can't calculate an answer to that question. In a Gilligan's Island economy, it's not really an issue of wanting the maximum utility for the maximum number, but rather that all the individuals want their own personal possible maximum. In other words, "All maxima are desired at once—by various participants."19 And in trying to fulfill their desires, every individual's actions will be influenced by expectations of everyone else's actions, and vice versa, the old "I think he thinks I think" problem. That makes a social economy, with multiple participants, inherently distinct from the Robinson Crusoe economy. "And it is this problem which the theory of ‘games of strategy' is mainly devised to meet," von Neumann and Morgenstern announced.20
Of course, it is easier said than done. It's one thing to realize that Gilligan's Island is more complex than Robinson Crusoe's; it's something else again to figure out how to do the math. Sure, you can start with something simple, like analyzing the interactions between just two people. Then, once you understand how two people will
interact, you can use the same principles to analyze what will happen when a third person enters the game, and then a fourth, and so on. (Eventually, then, you would possess the elusive Code of Nature, once you mastered the math of analyzing the behavior of all the individuals in society as a whole.)
However, you can see how things would rapidly become difficult to keep track of. Each person in the game (or the economy) will make choices based on a wide range of variables. In the Robinson Crusoe economy, his set of variables encompasses all those factors that would affect his quest for maximum utility. But if the Minnow had beached on Crusoe's island, each new player would have brought an additional set of variables of his or her own into the game. Then Crusoe would need to take all of those new variables into account, too.
On top of all that, more players means a more complex economy, more kinds of goods and services, different methods of production. So the social economy rapidly becomes a mathematical nightmare, it would seem, beyond even the ability of the know-it-all Professor to resolve. But there is hope, for economics and for understanding society, and it's a hope that's based on the simple idea of taking a temperature.
TAKING SOCIETY'S TEMPERATURE
In drawing analogies between economics and physics, von Neumann and Morgenstern talked a lot about the theory of heat (or, as it is more pretentiously known, thermodynamics). They pointed out, for instance, that measuring heat precisely did not lead to a theory of heat; physicists needed the theory first, in order to understand how to measure heat in an unambiguous way. In a similar way, game theory needed to be developed first to give economists the tools they needed to measure economic variables properly.
The example of the theory of heat played another crucial role—in articulating a basic issue within game theory itself. At the outset, von Neumann and Morgenstern made it clear that they did not want to venture into the philosophical quagmire of defining all the nuances of utility. For them, to develop game theory for use in economics, it was enough to equate utility with money. For the businessman, money (as in profits) is a logical measure of utility; for consumers, income (minus expenses) is a good measure of utility, or you could think of the utility of an object as the price you were willing to pay for it. And money can be used as a currency for translating what anyone wants into more specific objects or events or experiences or whatever. So equating utility with money is a convenient simplifying assumption, allowing the theory to focus on the strategic aspects of how to achieve what you want, without worrying about the complications involved in defining what you want.
However, there remained an important aspect of utility that von Neumann and Morgenstern had to address. Was it even possible, in the first place, to define utility in a numerical way, to make it susceptible to a mathematical theory? (Bernoulli had proposed a way to calculate utility, but he had not tried to prove that the concept could be a basis for making rational choices in a consistent way.) Money (which obviously is numerical) could really be a good stand-in for the more complex concept of utility only if utility can really be represented by a numerical concept. And so they had to show that it was possible to define utility in a mathematically rigorous way. That meant identifying axioms from which the notion of utility could be deduced and measured quantitatively.
As it turned out, utility could be quantified in a way not unlike the approach physicists used to construct a scientifically rigorous definition of temperature. After all, primitive notions of utility and temperature are similar. Utility, or preference, can be thought of as just a rank ordering. If you prefer A to B, and B to C, you surely prefer A to C. But it is not so obvious that you can ascribe a number to how much you prefer A to B, or B to C. It was once much the same with heat—you could say that something felt warmer or cooler than something else, but not necessarily how much, certainly not in a precise way—before the development of the theory of heat. But nowadays the absolute temperature scale, based on the laws of thermodynamics, gives temperature an exact quantitative meaning. And von Neumann and Morgenstern showed how you could similarly convert rank orderings into numerically precise measures of utility.
You can get the essence of the method from playing a modified version of Let's Make a Deal. (For the youngsters among you, that was a famous TV game show, in which host Monty Hall offered contestants a chance to trade their prizes for possibly more valuable prizes, at the risk of getting a clunker.) Suppose Monty offers you three choices: a BMW convertible, a top-of-the-line bigscreen plasma TV, or a used tricycle. Let's say you want the BMW most of all, and that you'd prefer the TV to the tricycle. So it's a simple matter to rank the relative utility of the three products. But here comes the deal. Your choice is to get either the plasma TV, OR a 50-50 chance of getting the BMW. That is, the TV is behind Door Number 1, and the BMW is behind either Door Number 2 or Door Number 3. The other door conceals the tricycle.
Now you really have to think. If you choose Door Number 1—the plasma TV—you must value it at more than 50 percent as much as the BMW. But suppose the game is more complicated, with more doors, and the odds change to a 60 percent chance of the BMW, or 70 percent. At some point you will be likely to opt for the chance to get the BMW, and at that point, you could conclude that the utilities are numerically equal—you value the plasma TV at, say, 75 percent as much as the BMW (plus 25 percent of the tricycle, to be technically precise). Consequently, to give utility a numerical value, you just have to arbitrarily assign some number to one choice, and then you can compare other choices to that one using the probabilistic version of Let's Make a Deal.
So far so good. But there remains the problem of operating in a social economy where your personal utility is not the only issue—you have to anticipate the choices of others. And in a small-scale Gilligan's Island economy, pure strategic choices can be subverted by things like coalitions among some of the players. Again, the theory of heat offers hope.
Temperature is a measure of how fast molecules are moving. In principle, it's not too hard to describe the velocity of a single molecule, just as you could easily calculate Robinson Crusoe's utility. But you'd have a hard time with Gilligan's Island, just as it becomes virtually impossible to keep track of all the speeds of a relatively few number of interacting molecules. But if you have a trillion trillion molecules or so, the interactions tend to average out, and using the theory of heat you can make precise predictions about temperature. (The math behind this is, of course, statistical mechanics, which will become even more central to the game theory story in later chapters.)
As von Neumann and Morgenstern pointed out, "very great numbers are often easier to handle than those of medium size."21 That was exactly the point made by Asimov's psychohistorians: Even though you can't track each individual molecule, you can predict the aggregate behavior of vast numbers, precisely what taking the temperature of a gas is all about. You can measure a value related to the average velocity of all the molecules, which reflects the way the individual molecules interact. Why not do the same for people? It worked for Hari Seldon. And it might work for a sufficiently large economy. "When the number of participants becomes really great," von Neumann and Morgenstern wrote, "some hope emerges that the influence of every particular participant will become negligible."22
With the basis for utility established at the outset, von Neumann and Morgenstern could proceed simply by taking money to be utility's measure. The bulk of their book was then devoted to the issue of finding the best strategy to make the most money.
At this point, it's important to clarify what they meant by the concept of strategy. A strategy in game theory is a very specific course of action, not a general approach to the game. It's not like tennis, for instance, where your strategy might be "play aggressively" or "play safe shots." A game theory strategy is a defined set of choices to make for every possible circumstance that might arise. In tennis, your strategy might be to "never rush the net when your opponent serves; serve and volley whenever you are even or ahead in a game; always stay back when
behind in a game." And you'd have other rules for all the other situations.
There's one additional essential point about strategy in game theory—the distinction between "pure" strategies and "mixed" strategies. In tennis, you might rush the net after every serve (a pure strategy) or you might rush the net after one out of every three serves, staying back at the baseline two times out of three (a mixed strategy). Mixed strategies often turn out to be essential for making game theory work.
In any event, the question isn't whether there is always a good general strategy, but whether there is always an optimum set of rules for strategic behavior that covers all eventualities. And in fact, there is—for two-person zero-sum games. You can find the best strategy using the minimax theorem that von Neumann published in 1928. His proof of that theorem was notoriously complicated. But its essence can be boiled down into something fairly easy to remember: When playing poker, sometimes you need to bluff.
MASTERING MINIMAX
The secret behind the minimax approach in two-person zero-sum games is the need to remember that whatever one player wins, the other loses (the definition of zero sum). So your strategy should seek to maximize your winnings, which would have the effect of minimizing your opponent's winnings. And of course your opponent wants to do the same.
Depending on the game, you may be able to play as well as possible and still not win anything, of course. The rules and stakes may be such that whoever plays first will always win, for instance, and if you go second, you're screwed. Still, it is likely that some strategies will lose more than others, so you would attempt to minimize your opponent's gains (and your losses). The question is, what strategy do you choose to do so? And should you stick with that strategy every time you play?
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